# Reduced Order Modeling using the Wavelet-Galerkin Approximation of Differential Equations

I started this work in the late fall of 2012 after working with the Finite Element Method(FEM) and its application to Reduced Order Modeling(ROM) approaches over the summer. After struggling for some time with the determination of the inner product calculations required for this type of work, I got my time dependent ROM code working. I defended this work in front of my Master of Science committee on October 30th, 2013 and was awarded my degree at the end of the semester.

The code used for this work is attached and is implemented in MATLAB. There is a sub-directory called "Nonlinear" that is not quite working yet as there seem to be some issues with the implementation of the boundary conditions (see the pdf entitled "Nonlinear_report"). This sub-directory contains code written in FORTRAN but I can't guarantee it is working properly.

The important files contained in this folder are:

**wavelet_ode.m:**Solves the partial differential equation with given set of input coefficients**generate_basis.m:**Solves the PDE with a number of different input parameters and then creates a set of reduced basis vectors based on these solutions**POD_simulate.m:**Uses the reduced basis vectors to create a reduced order solution to the PDE- A number of other functions/pictures/text files whose names are hopefully self-explanatory. If not please shoot me an email and I would be happy to explain myself!

## The Code

The code can be downloaded in a tar-zipped file here. And can be unzipped using the command: "tar xvfz Solve_DE.tar.gz"

## Abstract

Over the past few decades an increased interest in reduced order modeling approaches has led to its application in areas such as real time simulations and parameter studies among many others. In the context of this work reduced order modeling seeks to solve differential equations using substantially fewer degrees of freedom compared to a standard approach like the finite element method. The finite element method is a Galerkin method which typically uses piecewise polynomial functions to approximate the solution of a differential equation. Wavelet functions have recently become a relevant topic in the area of computational science due to their attractive properties including differentiability and multi-resolution. This research seeks to combine a wavelet-Galerkin method with a reduced order approach to approximate the solution to a differential equation with a given set of parameters. This work will focus on showing that using a reduced order approach in a wavelet-Galerkin setting is a viable option in determining a reduced order solution to a differential equation.

The remainder of my thesis can be downloaded here.